Homework 2
1
Calculus
Homework 2
Due Date: October 17 (Wednesday)
1. Find the slope and y-intercept (if possible) of the equation of the line.
(a) 5x + y = 20
(b) 2x + 3y = 9.
2. Find the equation of the line that passes through the given points.
(a) (4, 3), (0, -5)
(b) (-3, 6), (1, 2)
3. The graphs shows the revenue (in billions of dollars) for Verizon Communications for the
years 2003 through 2009.
(a) Determine the years in which the revenue increased the greatest and the least.
(b) Find the slope of the line segment connecting the points for the years 2003 and 2009.
(c) Interpret the meaning of the slope in part (b) in the context of the problem.
4. Your annual salary was $34,600 in 2009 and $37,800 in 2011. Assume your salary can be
modeled by a linear equation.
2
Calculus H415611 Fall 2012
(a) Write a linear equation giving your salary S (in dollars) in terms of the year t. Let
t = 9 represent 2009.
(b) Use the linear model to predict your salary in 2015.
5. Evaluate the difference quotient and simplify the result.
(a) f (x) = x2 − 5x + 2,
(b) f (x) =
f (x)−f (2)
√1 ,
x−2
x
f (x+4x)−f (x)
4x
=?
=?
6. Find the inverse function of f .
(a) f (x) = 23 x + 1, x ∈ R = (−∞, ∞)
√
(b) f (x) = x2 − 4, x ≥ 2.
7. The demand function for a commodity is
p=
14.75
,
1 + 0.01x
x≥0
where p is the price per unit and x is the number of units of sold.
(a) Find x as a function of p.
(b) Find the number of units sold when the price is $10.
8. The weekly cost C of producing x units in a manufacturing process is given by C(x) =
70x + 500. The number of units x produced in t hours is given by x(t) = 40t.
(a) Find and interpret C(x(t)).
(b) Find the cost of 4 hours of production.
(c) After how much time does the cost of production reach $18,000?
9. Find the limit (if it exists).
x2 − 9
x→−3 x + 3
2x2 − x − 3
(b) lim
x→−1
x+1
(t + 4t)2 − 4(t + 4t) + 2 − (t2 − 4t + 2)
(c) lim
4t→0
4t
(a) lim
10. Use a graph to find the limit from the left and the limit from the right.
|x − 6|
x→6
x−6
|x − 6|
(b) lim+
x→6
x−6
(a) lim−
11. The cost C (in thousands of dollars) of removing p% of the pollutants from the water in
a small lake is given by
25p
C=
, 0 ≤ p < 100.
100 − p
Homework 2
3
(a) Find the cost of removing 50% of the pollutants.
(b) What percent of the pollutants can be removed for $100 thousand?
(c) Evaluate lim − C. Explain your results.
p→100
12. Describe the interval(s) on which the function is continuous. Explain why the function is
continuous on the interval(s). If the function has a discontinuity, identify the conditions
of continuity that are not satisfied.
(a) f (x) = x2 − 2x + 1
(b) f (x) =
(c) f (x) =
x−1
x2 +x−2
√
3 + x, x ≤ 2;
x2 + 1, x > 2.
x2 − 4, x ≤ 0;
3x + 1, x > 0.
(d) f (x) =
(e) f (x) =
4−x
(f) f (x) =
|4−x|
4−x
(g) f (x) =
[[x]]
2
+x
(h) f (x) = x − [[x]].
(i) f (x) = [[2x]] + 1.
(j) h(x) = f (g(x)), f (x) =
1
,
x−1
g(x) = x2 + 5.
13. Find the constant a and b such that the function

x ≤ −1;
 2,
ax + b, −1 < x < 3;
f (x) =

−2,
x ≥ 3.
is continuous on the entire real number line.